Completeness in supergravity constructions

Communications in Mathematical Physics manuscript No.(will be inserted by the editor)

Completeness in supergravity constructions

V. Cortes1, X. Han1 and T. Mohaupt2

1 Department of Mathematics and Center for Mathematical Physics, University of Hamburg,

Bundesstraße 55, D-20146 Hamburg, Germany. E-mail: [email protected]

2 Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool,

Peach Street, Liverpool L69 7ZL, UK. E-mail: [email protected]

Received: date / Accepted: date

Abstract: We prove that the supergravity r- and c-maps preserve completeness.

As a consequence, any component H of a hypersurface h = 1 defined by a ho-

mogeneous cubic polynomial h such that −∂2h is a complete Riemannian metric

on H defines a complete projective special Kahler manifold and any complete

projective special Kahler manifold defines a complete quaternionic Kahler man-

ifold of negative scalar curvature. We classify all complete quaternionic Kahler

manifolds of dimension less or equal to 12 which are obtained in this way and

describe some complete examples in 16 dimensions.

1. Introduction

The supergravity r-map and the supergravity c-map are geometric constructions

known to theoretical physicists working in supergravity and string theory. They

can be obtained by dimensional reduction of the vector multiplet sector of su-

2 V. Cortes, X. Han and T. Mohaupt

pergravity theories with eight real supercharges from 5 to 4 and from 4 to 3

spacetime dimensions, respectively. The reduction from 4 to 3 dimensions was

worked out by Ferrara and Sabharwal [22], who used the resulting explicit local

description of the c-map to prove that it maps a projective special Kahler mani-

fold (see Definition 4) of real dimension 2n defined by a holomorphic prepotential

F = F (z0, . . . , zn) homogeneous of degree two to a quaternionic Kahler manifold

of dimension 4n + 4 of negative scalar curvature. Similarly, it was shown by de

Wit and Van Proeyen [21] that the r-map maps projective special real manifolds

(see Definition 1) of dimension n defined by a cubic polynomial h = h(x1, . . . , xn)

to projective special Kahler manifolds of dimension 2n + 2.

Despite some recent advances in the geometric understanding of the r-map

[3] and the c-map [29], as well as in finding new formulations of the c-map

within the formalisms of supergravity [16,18,19,36], very little is know about

global geometric properties of these constructions. In recent approaches to hy-

permultiplet moduli spaces, there have been efforts aiming at the computation

of quantum corrections to the Ferrara-Sabharwal metric inspired by the work

of Gaiotto, Moore and Neitzke on wall crossing [24]. While work along this

line has been successful in the limit where gravity decouples, that is in the

framework of hyper-Kahler geometry, part of the quantum corrections to the

Ferrara-Sabharwal metric have been computed using string dualities [25–27],

and the connection to wall crossing has been discussed in [28]. Despite these

encouraging results our understanding of the quantum corrected geometry of

hypermultiplets coupled to supergravity is still limited. In particular, the prob-

lem of completeness of quaternionic Kahler metrics obtained as deformations

Completeness in supergravity constructions 3

(e.g. by quantum corrections) of quaternionic Kahler metrics constructed by the

c-map is an interesting subject for future investigation.

The main results of this paper are Theorem 4 and Theorem 5, which state

that the supergravity r-map and the supergravity c-map preserve completeness.

As a consequence, we obtain an effective method for the construction of com-

plete projective special Kahler and quaternionic Kahler manifolds starting from

certain real affine hypersurfaces defined by homogeneous cubic polynomials h.

Any such hypersurface of dimension n defines a quaternionic Kahler manifold of

dimension 4n + 8, that is a point defines an 8-fold, a curve defines a 12-fold, a

surface defines a 16-fold etc. The study of the completeness of the quaternionic

Kahler manifold is reduced to that of the completeness of the cubic hypersur-

face H ⊂ Rn+1 with respect to the Riemannian metric −∂2h|H. We show how

to obtain interesting complete examples and even classification results in low

dimensions. In particular, we find two complete inhomogeneous examples, see

Corollary 4 b) and Example 3. The homogeneous examples are automatically

complete, as is any complete Riemannian manifold. Moreover, all known exam-

ples of homogeneous projective special Kahler manifolds with exception of the

complex hyperbolic spaces are in the image of the r-map and all known exam-

ples of homogeneous quaternionic Kahler manifolds of negative scalar curvature

with exception of the quaternionic hyperbolic spaces are in the image of the

c-map, see [21] and references therein. The known examples include the homo-

geneous projective special Kahler manifolds of semisimple groups classified in

[2] and the quaternionic Kahler manifolds admitting a simply transitive (com-

pletely) solvable group of isometries classified in [1,9]. The first class contains

only Hermitian symmetric spaces of noncompact type, whereas the second class


contains all quaternionic Kahler symmetric spaces of noncompact type as well

as all known nonsymmetric homogeneous quaternionic Kahler manifolds. We

plan to work towards general completeness results for hypersurfaces in higher

dimensions in the future.

One of the inhomogeneous complete examples is the ‘quantum STU model’

[6,5,31,30]. This model, or actually family of models, can be constructed by

compactification of the heterotic string on manifolds with holonomy contained

in SU(2) and with instanton numbers (12, 12) or (13, 11) or (10, 4). The special

real, special Kahler and special quaternionic manifold occur as moduli spaces of

compactifications on K3 × S1, K3 × T 2 and K3 × T 3, respectively. The qual-

ification ‘quantum’ refers to the quantum corrections to the cubic part of the

underlying Hesse potential

h = STU → h = STU +13U3 ,

which makes the corresponding manifolds inhomogeneous. This deformation is

captured by the triple intersection forms of dual models, which are compactifica-

tions on Calabi-Yau threefolds which are elliptic fibrations over the Hirzeburch

surfaces F0, F1, F2, respectively [31,34,40]. The choice of the base of the fibra-

tion corresponds to the choice of instanton numbers in the dual heterotic model.

The heterotic compactifications on K3 × T 2 are dual to type-IIA compactifi-

cations on the corresponding Calabi-Yau threefolds, and the vector multiplet

moduli spaces of these models are the complexified Kahler cones of the Calabi-

Yau threefolds. The type-IIA model has an ‘M-theory lift’ to a compactification

of eleven-dimensional M-theory on the same Calabi-Yau threefold, which is then

dual to the heterotic compactification on K3 × S1. The moduli spaces of five-

dimensional vector multiplets correspond to a fixed volume slice of the (real)


Kahler cones of the underlying Calabi-Yau threefolds. Some properties of the

special Kahler and special real metrics occuring in the vector multiplet sectors

of these models have been discussed in the physics literature [32,7,33]. The

geodesically complete spaces considered in this paper are the natural choices

of scalar manifolds if these models are considered as supergravity models. The

moduli spaces relevant for string theory are sub-domains of these manifolds, as

discussed in [34,40,32,7,33].

In the last section we give a geometric interpretation of the complex (n +

1)× (n + 1)-matrix N = (NIJ ), which defines the nontrivial part of the Ferrara-

Sabharwal metric. We show that N defines a Weil flag which is precisely the

image of the Griffiths flag associated with the variation of Hodge structure of

weight 3 defined by the underlying affine special Kahler manifold under a natural

Sp(R2n+2)-equivariant map from Griffiths to Weil flags, see Corollary 5. Further-

more, N is canonically associated with a positive definite metric which differs

from the (indefinite) affine special Kahler metric by a canonical sign switch, see

Corollary 6. Using this geometric insight, we are able to extend the c-map and

our completeness result to special Kahler manifolds which do not admit a global

description by a single prepotential, see Theorem 10.

2. Completeness of metrics on product manifolds and bundles

Let us recall that a Riemannian manifold (M, g) is called complete if it is com-

plete as a metric space, i.e. if every Cauchy sequence in M converges. The basic

result in Riemannian geometry concerning completeness is the following theorem

of Hopf and Rinow, cf. [35] Ch. 5, Thm. 21.


Theorem 1 (Hopf-Rinow) For a Riemannian manifold (M, g) the following

conditions are equivalent:

(i) M is complete.

(ii) M is geodesically complete, i.e. every inextendible geodesic ray has infinite

length.

(iii) Any closed and bounded subset of M is compact.

We give another equivalent formulation of completeness in terms of (smooth)

curves γ : I → M , where I ⊂ R is a (nondegenerate) interval.

Lemma 1 A Riemannian manifold (M, g) is complete if and only if every

curve γ : I → M which is not contained in any compact subset of M has infinite

length.

Proof: “⇒” Let us assume that M is complete and that γ : I → M is not

contained in any compact set. Then γ(I) ⊂ M is unbounded, since it is not

contained in any ball. Here we use the fact that closed balls are compact in any

complete Riemannian manifold, by Theorem 1 (iii). Clearly, an unbounded curve

has infinite length.

“⇐” If M is not complete, then there exists an inextendible geodesic ray γ :

[0, L) → M of finite length L, by Theorem 1 (ii). The ray γ is not contained in

any compact set K, because otherwise γ(t) would converge to a limit point in

K for t → L and γ could then be extended to a geodesic ray γ : [0, L) → M for

some L > L. ⊓⊔

Let M = M1 × M2 be a product manifold and denote by πi : M → Mi,

i = 1, 2, the projections. We consider Riemannian metrics g of the form

g = g1 + g2, (2.1)


where g1 is (the pullback of) a Riemannian metric on M1 and g2 ∈ Γ (π∗2S2T ∗M2)

is a family of Riemannian metrics on M2 depending on a parameter p ∈ M1.

Notice that the tensors g1 and g2 take the form

g1 =∑

g(1)ab (x)dxadxb, g2 =

∑g(2)αβ (x, y)dyαdyβ ,

with respect to local coordinates x = (xa) on M1 and y = (yα) on M2.

We will assume that the tensor field g2 satisfies the following condition:

(C) For all compact subsets A ⊂ M1 there exists a complete Riemannian metric

gA on M2 such that g2 ≥ π∗2gA on A × M2 ⊂ M .

Lemma 2 Assume that (M1, g1) is complete and that g2 satisfies the condition

(C). Then (M, g) is complete.

Proof: Let γ = (γ1, γ2) : I → M = M1 × M2 be a curve which is not contained

in any compact subset K ⊂ M . In view of Lemma 1, it suffices to show that any

such curve γ has infinite length. From the assumption on γ it follows that

(i) γ1 : I → M1 is not contained in any compact subset K1 ⊂ M1 or

(ii) γ2 : I → M2 is not contained in any compact subset K2 ⊂ M2.

(Otherwise, γ would be contained in a compact set K = K1 × K2.) In case (i),

L(γ1) = ∞, by the completeness of (M1, g1) and Lemma 1. Comparing lengths

we obtain

L(γ) ≥ L(γ1) = ∞

and, hence, L(γ) = ∞. If (i) is not satisfied, then γ1(I) is contained in a compact

set A = K1 ⊂ M1. By the completeness of (M2, gA) and property (ii), we

now have that LgA(γ2) = ∞. Now it suffices to compare the lengths: L(γ) ≥

LgA(γ2) = ∞. ⊓⊔


Theorem 2 Let (M1, g1) be a complete Riemannian manifold and (g2(p))p a

smooth family of G-invariant Riemannian metrics on a homogeneous manifold

M2 = G/K, depending on a parameter p ∈ M1. Then the Riemannian metric

g = g1 + g2 on M = M1 × M2 is complete. Moreover, the action of G on M2

induces an isometric action of G on (M, g).

Proof: The last assertion is obvious. For the completeness of (M, g), it suffices

to check that g2 satisfies the condition (C) of Lemma 2. We use the natural one-

to-one correspondence between G-invariant Riemannian metrics on M2 = G/K

and K-invariant scalar products on the vector space ToM2∼= g/k, o = eK. Under

this correspondence, the family (g2(p))p corresponds to a family (β(p))p of scalar

products. For every compact subset A ⊂ M1 the family (β(p))p∈A is uniformly

bounded from below by a scalar product βA:

β(p) ≥ βA, for all p ∈ A.

This implies

g2(p) ≥ gA, for all p ∈ A,

for the G-invariant Riemannian metric gA associated with βA. Now it suffices to

remark that gA is complete, as is any G-invariant Riemannian metric on G/K.

⊓⊔

Corollary 1 Let gU =∑

gabdxadxb be a complete Riemannian metric on a

domain U ⊂ Rn. Then the metric

gM =34

∑gab(x)(dxadxb + dyadyb) (2.2)

on M = U×Rn is complete. The action of Rn by translations in the y-coordinates

is isometric and the projection M → U is a principal fiber bundle with structure

group Rn. The submanifold U = U × 0 ⊂ U × Rn = M is totally geodesic.


(The factor 34 is introduced only in order to obtain the usual normalization of

the projective special Kahler metric for the r-map defined in the next section.)

Proof: For the completeness it suffices to apply Theorem 2 to the case M2 =

G = Rn, g2 =∑

gab(x)dyadyb. U ⊂ M is totally geodesic as the fixed point set

of the isometric involution (x, y) 7→ (x,−y). ⊓⊔

2.1. Generalisation to the case of bundles . More generally, for later applications

we consider now a bundle π : M → M1 with standard fiber M2 over a Rieman-

nian manifold (M1, g1). We suppose that the total space M is endowed with

a Riemannian metric g such that for all p ∈ M1 there exists a neighbourhood

U ⊂ M1 and a local trivialisation π−1(U) ∼= U × M2 with respect to which the

metric takes the form

g|π−1(U) = g1|U + gU2 , (2.3)

where gU2 is a smooth family of Riemannian metrics on M2 depending on a

parameter in U . Such metrics g will be called bundle metrics. We will assume

that gU2 satisfies the condition (C) for all compact subsets A ⊂ U . Lemma 2 and

Theorem 2 have the following straightforward generalisations:

Lemma 3 Assume that (M1, g1) is complete and that the local fiber metrics

gU2 in equation (2.3) satisfy the condition (C) for all compact subsets A ⊂ U .

Then the bundle metric g is complete.

Theorem 3 Let g be a bundle metric on a bundle π : M → M1 over a

complete Riemannian manifold (M1, g1) and assume that the standard fiber is a

homogeneous space M2 = G/K and that the fiber metrics gU2 are G-invariant.

Then (M, g) is complete.


3. The generalized r-map preserves completeness

Let U ⊂ Rn be a domain which is invariant under multiplication by positive

numbers and let h : U → R>0 be a smooth function which is homogeneous of

degree d ∈ R \ 0, 1. Then

H := x ∈ U |h(x) = 1 ⊂ U

is a smooth hypersurface and U = R>0 · H ∼= R>0 × H. We will assume that

− 1d∂2h is positive definite on TH. This easily implies that − 1

d∂2h is a Lorentzian

(if d > 1) or Riemannian (if d < 1) metric on U which restricts to a Riemannian

metric gH on H.

Definition 1 The Riemannian manifold (H, gH) is called a projective special

real manifold if, in addition, h is a polynomial function of degree d = 3.

Example Let V = H1,1(X, R) be the (1, 1)-cohomology of a compact Kahler

manifold X of complex dimension d ≥ 2 and U ⊂ V the Kahler cone. We define

a homogeneous polynomial h of degree d on V by

h(a) = a∪d = a ∪ · · · ∪ a, a ∈ V.

The polynomial h defines a positive function on U and the metric gH defined

above is positive definite on the hypersurface

H := x ∈ U |h(x) = 1 ⊂ V ∼= Rn, n = h1,1(X).

This follows from the Hodge-Riemann bilinear relations, see [39] Chap. V, Sec. 6,

which imply that gH is positive definite on the primitive cohomology H1,10 (X, R) =

TκH defined as the kernel of the cup product with κd−1 : H2(X, R) → H2d(X, R),

where κ ∈ U is a Kahler class. If d = 3 then (H, gH) is a projective special real

manifold.


We endow U with the Riemannian metric

gU = −1d∂2 ln h (3.1)

and M = TU ∼= U × Rn with the metric (2.2).

Proposition 1 (U, gU ) is isometric to the product (R × H, dr2 + gH).

Proof: This is a straightforward calculation using the diffeomorphism

R × H → U, (r, x) 7→ erx,

and the formula (3.1). ⊓⊔

Definition 2 The correspondence (H, gH) 7→ (M, gM ) is called the generalized

r-map. The restriction to polynomial functions h of degree d = 3 is called the

supergravity r-map.

Next we need to recall the notion of a projective special Kahler manifold [4,

20,23]. It is best explained starting from the notion of a conical special Kahler

manifold, cf. [4,12]. For the purpose of this paper, we shall restrict the signature

of the metric by the condition (iv) in the following definition.

Definition 3 A conical special Kahler manifold (M,J, g,∇, ξ) is a pseudo-

Kahler manifold (M,J, g) endowed with a flat torsionfree connection ∇ and a

vector field ξ such that

(i) ∇ω = 0, where ω = g(·, J ·) is the Kahler form,

(ii) d∇J = 0, where J is considered as a one-form with values in the tangent

bundle,

(iii) ∇ξ = Dξ = Id, where D is the Levi-Civita connection,

(iv) g is positive definite on the plane D = spanξ, Jξ and negative definite on

D⊥.


It is shown in [4,12] that the geometric data of a conical special Kahler manifold

can be locally encoded in a holomorphic function F homogeneous of degree 2

defined in a domain MF ⊂ Cn, see Theorem 2 and Proposition 6 [12]. In fact, F

is the generating function of a holomorphic Lagrangian immersion MF → C2n,

which induces on MF the structure of a conical special Kahler manifold. F is

called the holomorphic prepotential in the supergravity literature [20]. Under the

assumptions of Definition 3, the vector fields ξ and Jξ define a holomorphic

action of a two-dimensional Abelian Lie algebra. We will assume that this in-

finitesimal action lifts to a principal C∗-action on M with the base manifold

M = M/C∗. Then the hypersurface S := p ∈ M |g(ξ(p), ξ(p)) = 1 ⊂ M is an

S1-principal bundle over M . The principal action is isometric, since it is gener-

ated by the Killing vector field Jξ. Therefore, the Lorentzian metric gS = −g|S

induces a Riemannian metric g on M , which is easily seen to be Kahlerian. In

fact, the negative definite Kahler manifold (M,−g) is precisely the Kahler re-

duction of (M, g) with respect to the above isometric Hamiltonian S1-action for

a positive level of the moment map, which is µ = g(ξ,ξ)2 .

Definition 4 The Kahler manifold (M, g) is called a projective special Kahler

manifold. The metric g is called a projective special Kahler metric.

A standard example of a projective special Kahler metric is the Weil-Petersson

metric on the space of complex structure deformations of a Calabi-Yau 3-fold,

that is of a compact Kahler manifold with holonomy SU(3), see, for instance,

[37,10].

Theorem 4 The generalized r-map maps complete Riemannian manifolds

(H, gH) as above to complete Kahler manifolds (M, gM ) with a free isometric


action of the vector group Rn. The supergravity r-map maps complete projective

special real manifolds to complete projective special Kahler manifolds.

Proof: The isometric action of Rn exists for general metrics as in Corollary 1.

The Kahler property follows from the fact that the metrics gU considered here

are of Hessian type, cf. [3] Prop. 3. In fact, (H, gH) is mapped to M = U ×Rn ∼=

Rn + iU ⊂ Cn with the metric gM defined by the Kahler potential − lnh(x),

where (x, y) ∈ U ×Rn = M is identified with ζ = y + ix ∈ Rn + iU ⊂ Cn. If h is

a cubic polynomial then a simple calculation shows that the metric gM can be

also obtained from the Kahler potential K(1, ζ) defined by

K(z) = − ln

(i

n∑I=0

(zI FI − FI z

I))

,

where

F (z0, . . . , zn) = h(z1, . . . , zn)/z0 (3.2)

is a holomorphic function homogeneous of degree 2 on the domain

M := z0 · (1, ζ)|z0 ∈ C∗, ζ ∈ Rn + iU ⊂ Cn+1.

(It suffices to check that i|z0|2

∑(zI FI−FI z

I) = 8h(x).) This shows that (M, gM )

is a projective special Kahler manifold with the holomorphic prepotential F .

The corresponding conical special Kahler manifold is the C∗-bundle M → M

endowed with the affine special Kahler metric gM = 2∑n

I,J=0(ImFIJ)dzIdzJ ,

which has signature (2, 2n). By Proposition 1 and Corollary 1, the completeness

of (H, gH) implies that of (U, gU ) and the completeness of (U, gU ) implies that

of (M, gM ). ⊓⊔

Corollary 2 Any complete projective special real manifold (H, gH) of dimen-

sion n−1 ≥ 0 admits a canonical realisation as a totally geodesic submanifold of


a complete projective special Kahler manifold (M, gM ) with a free isometric ac-

tion of the group Rn. Each orbit of Rn is flat and and intersects the submanifold

U = R>0 · H ⊂ M orthogonally in exactly one point.

Proof: The orbits are flat since the metric is translation invariant. The hyper-

surface H ⊂ U ∼= R>0 × H is totally geodesic in virtue of Proposition 1 and

U ⊂ M is totally geodesic in virtue of Corollary 1. Therefore H ⊂ M is totally

geodesic. ⊓⊔

Notice that the above proof shows that H ⊂ M = r(H) is totally geodesic

also for the generalized r-map.

4. The supergravity c-map preserves completeness

A projective special Kahler manifold (M, gM ) which is globally defined by a

single holomorphic prepotential F is called a projective special Kahler domain.

Notice that the manifolds in the image of the r-map are defined by the pre-

potential (3.2) and, hence, are examples of projective special Kahler domains.

Recall [22] that the supergravity c-map maps projective special Kahler domains

(M, gM ) of dimension 2n to quaternionic Kahler manifolds (N, gN ) of dimension

4n + 4 and of negative scalar curvature. More generally, we may consider the

situation when the projective special Kahler manifold (M, gM ) is covered by a

collection of charts Uα on which the special Kahler geometry is encoded by a

prepotential Fα. The investigation of this case is postponed to section 6. For a

projective special Kahler domain (M, gM ), the quaternionic Kahler metric gN


on N = M × R2n+3 × R>0 ∼= M × R2n+4 is of the form

gN = gM + gG, (4.1)

gG =1

4ϕ2dϕ2 +

14ϕ2

(dϕ +∑

(ζIdζI − ζIdζI))2 +12ϕ

∑IIJ(p)dζIdζJ

+12ϕ

∑IIJ(p)(dζI +

∑RIK(p)dζK)(dζJ +

∑RJL(p)dζL), (4.2)

where (ζI , ζI , ϕ, ϕ), I = 0, 1, . . . , n, are coordinates on R2n+4 ⊃ R2n+3×R>0 and

the metric is defined for ϕ > 0. The matrices (IIJ(p)) and (RIJ(p)) depend only

on p ∈ M and (IIJ(p)) is invertible with the inverse (IIJ (p)). More precisely,

NIJ := RIJ + iIIJ := FIJ + i

∑K NIKzK

∑L NJLzL∑

IJ NIJzIzJ, NIJ := 2ImFIJ , (4.3)

where F is the holomorphic prepotential with respect to some system of special

holomorphic coordinates zI on the underlying conical special Kahler manifold

M → M . Notice that the expressions are homogeneous of degree zero and, hence,

well defined local functions on M . Also note that our conventions are slightly

different from those in [22]. The prepotentials are related by F = i4F [FS], while

N = N [FS]. Also note that I = −R[FS], and therefore I is positive definite.

Let (M, gM ) be a special Kahler domain and N = M × G the corresponding

quaternionic Kahler manifold with the Ferrara-Sabharwal metric gN = gM +gG.

We will show that, for fixed p ∈ M , gG(p) can be considered as a left-invariant

Riemannian metric on a certain Lie group diffeomorphic to R2n+4. We define

the Lie group G by putting the following group multiplication on R2n+4:

(ζ, ζ, ϕ, ϕ) · (ζ ′, ζ ′, ϕ′, ϕ′)

:= (ζ + eϕ/2ζ ′, ζ + eϕ/2ζ ′, ϕ + eϕϕ′ + eϕ/2(ζtζ ′ − ζ ′tζ), ϕ + ϕ′),

where ζt = (ζ0, . . . , ζn) is the transposed of the column vector ζ. We remark that

G is isomorphic to the solvable Iwasawa subgroup of SU(1, n + 2), which acts


simply transitively on the complex hyperbolic space of complex dimension n+2.

G is a rank one solvable extension of the (2n+3)-dimensional Heisenberg group.

We can realise it as a group of affine transformations of R2n+4 by associating to

an element (v, v, α, λ) ∈ G = R2n+4 the affine transformation

(ζ, ζ, ϕ, ϕ) 7→ (eλ/2ζ + v, eλ/2ζ + v, eλ/2(vtζ − vtζ) + eλϕ + α, eλϕ). (4.4)

In virtue of this action, we can identify G with the orbit of the point (0, 0, 1, 0),

which is L = R2n+3 × R>0 ⊂ R2n+4. This identification is simply

G ∋ (v, v, α, λ) 7→ (v, v, α, eλ) ∈ L.

One can easily check that the following pointwise linearly independent one-forms

are invariant under the transformations (4.4) and, hence, define a left-invariant

coframe on G ∼= L:

ηI :=√

2ϕ

dζI , ξI :=√

2ϕ

(dζI +

∑RIK(p)dζK

), (4.5)

ξn+1 :=dϕ

ϕ, ηn+1 :=

1ϕ

(dϕ +

∑(ζIdζI − ζIdζI)

).

This shows, in particular, that the metric gG(p) is left-invariant for all p ∈ M .

Theorem 5 The supergravity c-map maps any complete projective special

Kahler domain (M, gM ) to a complete quaternionic Kahler manifold (N, gN )

of negative scalar curvature, which admits the free isometric action (4.4) of the

group G.

Proof: The above description of the c-map, starting from (4.1), shows that the

quaternionic Kahler manifold (N, gN ) is of the form N = M ×G, gN = gM +gG,

where gG is a smooth family of left-invariant metrics on the group G depending

on a parameter p ∈ M . Therefore, Theorem 2 shows that (N, gN ) is complete if

(M, gM ) is complete. ⊓⊔


Corollary 3 Any complete projective special Kahler domain (M, gM ) admits

a canonical realisation as a totally geodesic Kahler submanifold of a complete

quaternionic Kahler manifold (N, gN ) with a free isometric action of the group

G. Each orbit of G is isometric to a complex hyperbolic space of holomorphic

sectional curvature −4 and intersects the submanifold M ⊂ N orthogonally in

exactly one point.

Proof: The submanifold M = M × e ⊂ M × G = N of the quaternionic

Kahler manifold N defined by the supergravity c-map is the fixed point set of

the isometric involution

(ζI , ζI , ϕ, ϕ) 7→ (−ζI ,−ζI ,−ϕ, ϕ−1).

This implies that M is totally geodesic. Next we compare gG(p) with the stan-

dard left-invariant Kahler metric of constant holomorphic sectional curvature −1

on G, which originates from the simply transitive action of G on the complex

hyperbolic space. By a linear change of the coordinates ζI we may assume that

the positive definite symmetric matrix IIJ(p) = δIJ . Then

gG(p) =14

n+1∑i=0

(ξ2i + (ηi)2),

where the left-invariant coframe (4.6) has the following differentials

dξn+1 = 0, dηn+1 = −n+1∑I=0

ξI ∧ηI , dξI = −12ξn+1∧ξI , dηI = −1

2ξn+1∧ηI .

This shows that, up to multiplication with the factor 1/2, the above coframe is

dual to the standard orthonormal basis of the elementary Kahlerian Lie algebra

g and, hence, that gG(p) is a Kahler metric of constant holomorphic sectional

curvature −4. ⊓⊔


5. Examples: Complete quaternionic Kahler manifolds associated to

cubic polynomials

As an immediate corollary of Theorem 4 and Theorem 5 we have:

Theorem 6 To any complete projective special real manifold (H, gH) the com-

position of the r-map with the c-map associates a complete quaternionic Kahler

manifold (N, gN ) of negative scalar curvature.

Example We can consider the point H = 1 ⊂ x3 = 1 ⊂ R as an example

of a projective special real manifold. The corresponding complete quaternionic

Kahler eightfold obtained by the above construction is the symmetric space

G∗2/SO(4) of noncompact type.

A homogeneous cubic polynomial h ∈ S3(Rn)∗ will be called hyperbolic (re-

spectively, elliptic) if there exists a point p ∈ h = 1 := x ∈ Rn|h(x) = 1 such

that ∂2h is negative (respectively, positive) definite on the tangent space of the

hypersurface h = 1 at p. Such points will be called hyperbolic (respectively,

elliptic). Let us denote by H = H(h) ⊂ h = 1 the open subset of hyperbolic

points. It is a projective special real manifold with the metric gH given by the

restriction of −13∂2h.

The classification of complete projective special real manifolds reduces to the

following two problems:

Problem 1 Classify all hyperbolic homogeneous cubic polynomials up to lin-

ear transformations. In other words, describe the orbit space S3(Rn)∗hyp/GL(n),

where S3(Rn)∗hyp ⊂ S3(Rn)∗ stands for the open subset of hyperbolic polynomials.


Problem 2 For each hyperbolic homogeneous cubic polynomial h, determine

the components of the hypersurface H(h) which are complete and classify them

up to linear transformations.

We will solve these problems in the simplest case, that is for n = 2. This gives

the classification of complete projective special real curves. The classification of

complete projective special real surfaces is open, but we will give some examples

of such surfaces.

5.1. Classification of complete cubic curves and corresponding 12-dimensional

quaternionic Kahler manifolds.

Theorem 7 The orbit space S3(R2)∗hyp/GL(2) consists of three points, which

are represented by the polynomials

x2y, x(x2 − y2) and x(x2 + y2).

Proof: Let h ∈ S3(R2)∗hyp. Interchanging the variables x and y, if necessary, we

can assume that degx h, the degree of h = h(x, y) in the variable x, is 2 or 3.

Since the Hessian of h is nondegenerate we also have degy h ≥ 1.

Case 1) If degx h = 3, then the polynomial f(x) := h(x, 1) has degree 3. Any

polynomial f(x) of degree 3 can be brough to one of the following forms by an

affine transformation in the variable x:

x3, cx2(x − 1), c(x + a)x(x − 1), cx(x2 + 1), c(x + a)(x2 + 1),

where c = 0 and a > 0 are real constants. The first form is excluded, since

degy h ≥ 1. This implies that h can be brought to one of the following forms by

a linear transformation:

x2(x − y), (x + ay)x(x − y), x(x2 + y2), (x + ay)(x2 + y2), a > 0.


The first polynomial is linearly equivalent to x2y. The zero set of the second

polynomial in the real projective line consists of the points −a, 0, 1 ∈ R ⊂

RP 1 = P (R2). Since any three pairwise distinct points in the projective line

are related by an element of GL(2), we can assume that a = 1. Finally, the

last polynomial can be brought to the form x(x2 + y2) by a linear conformal

transformation. Therefore, we are left with the following 3 normal forms:

x2y, x(x2 − y2) and x(x2 + y2). (5.1)

Case 2) If degx h = 2, then the quadratic polynomial f(x) = h(x, 1) can be

brought to one of the following forms by an affine transformation:

±x2, cx(x − 1), c(x2 + 1), c = 0.

Therefore h can be brought to one of the following forms by a linear transfor-

mation:

x2y, xy(x − y), (x2 + y2)y.

The last two polynomials are equivalent to the polynomials x(x2 ∓ y2) already

included in our list (5.1).

It remains to check that all 3 polynomials are indeed hyperbolic. For dimen-

sional reasons (n = 2), this is equivalent to the existence of a point p = (x, y) ∈

R2 such that h(p) > 0 and D(p) := det ∂2h(p) < 0.

a) For h = x2y we have D = −4x2. Therefore all points of the curve h = 1

are hyperbolic.

b) The same is true for h = x(x2 − y2), since D = −4(3x2 + y2).

c) For h = x(x2 + y2) we find D = 4(3x2 − y2). The point 13√5

(1, 2) ∈ h = 1

is hyperbolic, whereas 13√2

(1, 1) ∈ h = 1 is elliptic.


⊓⊔

Next we investigate the components of H(h).


Theorem 8

a) The curve x2y = 1 consists of hyperbolic points and has two equivalent

components. They are homogeneous and, hence, complete projective special

real curves.

b) The curve x(x2 − y2) = 1 consists of hyperbolic points and has three equiv-

alent components, which are inhomogeneous complete projective special real

curves.

c) Let h = x(x2 + y2). The curve H(h) = p ∈ R2|h(p) = 1, D(p) < 0, which

consists of the hyperbolic points of h = 1 has two equivalent components.

They are incomplete. The curve p ∈ R2|h(p) = 1, D(p) > 0 which consists

of the elliptic points of h = 1 is connected and incomplete.

Proof: a) h = x2y. The reflection x 7→ −x interchanges the two components

of the curve h = 1 and the subgroup diag(λ, λ−2)|λ > 0 ⊂ GL(2) acts

transitively on each component.

b) h = x(x2 − y2). The transformation−1/2 1/2

−3/2 −1/2

∈ SL(2)

generates a cyclic group of order three, which interchanges the three components

of the curve h = 1. Let us consider the component

C := p ∈ R2|h(p) = 1, x > 0.

It is symmetric with respect to the x-axis, intersects the x-axis at x = 1 and

approaches the asymptotic lines y = ±x when x → ∞. To prove the completeness

of C, it suffices to show that the length of the upper half C+ = C ∩ y > 0 of


the curve is infinite. A straighforward calculation shows that the metric ds2 =

g = − 13∂2h|C is given by the formula

32g = −3xdx2 + xdy2 + 2ydxdy =

3(4x3 − 1)4x2(x3 − 1)

dx2.

This yields the following asymptotics

g

dx2=

2x2

+ O

(1x5

)ds

dx=

√2

x+ O

(1x4

),

which implies that the arc length∫ x

1ds =

√2 lnx+O( 1

x3 ) grows logarithmically

with x. This shows that C+ has infinite length with respect to g. A simple

calculation shows that the automorphism group of C is trivial.

c) h = x(x2 + y2). The two components of H(h) = h = 1, x < 13√4

are

interchanged by the reflection y 7→ −y. They are incomplete due to the points

of inflections at the boundary points ( 13√4

,±√

33√4

). The same is true for the curve

p ∈ R2|h(p) = 1, D(p) > 0 = h = 1, x > 13√4

. ⊓⊔

Corollary 4 There exists precisely two complete projective special real curves,

up to linear equivalence:

a) (x, y) ∈ R2|x2y = 1, x > 0 and

b) (x, y) ∈ R2|x(x2 − y2) = 1, x > 0.

Under the composition of the r- and c-map both give rise to complete quaternionic

Kahler manifolds of dimension 12. In the first case we obtain the symmetric

quaternionic Kahler manifold

SO0(4, 3)SO(4) × SO(3)

.

The second example gives rise to a new complete quaternionic Kahler manifold.


5.2. Examples of complete cubic surfaces and corresponding 16-dimensional qua-

ternionic Kahler manifolds. Example 1 (STU model) The surface H = xyz =

1, x > 0, y > 0 ⊂ R3 is a homogeneous projective special real manifold. In

fact, the group R>0 × R>0 acts simply transitively on H by unimodular diago-

nal matrices. The corresponding quaternionic Kahler manifold is the symmetric

space

SO0(4, 4)SO(4) × SO(4)

.

Example 2 The surface H = x(xy − z2) = 1, x > 0 ⊂ R3 is another homo-

geneous projective special real manifold. It admits the following simply tran-

sitive solvable group of linear automorphisms: (x, y, z) 7→ (λ−2x, λ4(µx + y +

2µz), λ(µx+z)), λ > 0, µ ∈ R. The corresponding quaternionic Kahler manifold

is the nonsymmetric homogeneous manifold T(1) described in [9].

Example 3 (quantum STU model) The surface H = x(yz+x2) = 1, x < 0, y > 0

is an inhomogeneous complete projective special real manifold. Its automor-

phism group is one-dimensional. The maximal connected subgroup is given by:

(x, y, z) 7→ (x, λy, λ−1z), λ > 0. Under the r-map, the surface H gives rise to

a new complete projective special Kahler manifold, which is mapped to a new

complete quaternionic Kahler manifold of dimension 16 under the c-map. Let us

check the completeness of H. A straightforward calculation shows that:

−∂2h|H = 2(1 − x3)(

dx2

x2+

dy2

y2

)+ 2(1 + 2x3)

dxdy

xy

≥ 2(

1 − x3 − |1 + 2x3|2

)(dx2

x2+

dy2

y2

)≥ dx2

x2+

dy2

y2.


So the projective special real metric g = −13∂2h|H is bounded from below by

the product metric

dx2

3x2+

dy2

3y2

on R<0 × R>0, which is complete. In fact, dx2

3x2 + dy2

3y2 = dx2 + dy2 under the

change of variables x = 1√3

ln(−x), y = 1√3

ln y, which maps R<0 × R>0 to R2.

6. Globalisation of the Ferrara-Sabharwal metric

In this section we will investigate the problem of gluing Ferrara-Sabharwal

manifolds (Nα = Mα × G, gα = gMα + gαG) obtained from projective special

Kahler domains Mα ⊂ M in a projective special Kahler manifold (M, gM )

to a global quaternionic Kahler manifold (N, gN ). Here gMα = gM |Mα . Re-

call that G = R2n+4 with the group structure defined in section 4. Denote

by Mα = π−1M (Mα) ⊂ M the corresponding special coordinate domain in the

underlying conical special Kahler manifold πM : M → M . The affine special

coordinates on Mα will be denoted by qa, or, more precisely, by qaα. The holo-

morphic special coordinates will be zI or zIα. Let (Mα)α be a covering of M by

projective special Kahler domains. Then we define the quotient

N =∪α

Nα/ ∼

by the equivalence relation

Nα ∋ (m, v) ∼ (m′, v′) ∈ Nβ :⇐⇒ m = m′ and v = Aαβv′,

where Aαβ is the linear symplectic transformation such that qα = Aαβqβ and

A = diag((AT )−1,12).


Theorem 9 The natural projection π : N → M is a symplectic vector bun-

dle and at the same time a bundle of Lie groups. Each fiber is isomorphic to

the solvable Lie group G. There exists a unique quaternionic Kahler structure

(Q, gN ) on N such that gN |Nα = gα. Up to an isomorphism of quaternionic

Kahler manifolds consistent with the bundle structures, the quaternionic Kahler

manifold (N,Q, gN ) does neither depend on the covering (Mα)α of M nor on

the choice of special coordinates on the domains Mα.

Proof: The transition functions (Aαβ) can be considered as a Cech 1-cocycle

with values in the group

Sp(2n + 2, R) → Sp(2n + 4, R),

which defines the structure of a symplectic vector bundle on N . The above linear

action of Sp(2n + 2, R) on G = R2n+4 is by automorphisms of the solvable Lie

group G, which means that the gluing preserves the group structure of the fibers.

In order to prove that the local metrics gα can be glued to a global Riemannian

metric gN it suffices to check that gβG = A∗gα

G, since we know already that

gMα = gMβon Mα ∩ Mβ . It is useful to rewrite gG = gα

G in the following way:

gG =1

4ϕ2dϕ2 +

14ϕ2

(dϕ +∑

paΩabdpb)2 +12ϕ

∑Habdpadpb . (6.1)

Here (pa) = (ζI , ζJ),

Ω−1 = (Ωab) :=

0 −1n+1

1n+1 0

, (6.2)

H−1 = (Hab) :=

I−1 I−1R

RI−1 I + RI−1R

,


that is

H = (Hab) =

I + RI−1R −RI−1

−I−1R I−1

. (6.3)

We observe that 2Ω is the matrix representing the Kahler form ω = 2∑

dxI∧dyI

of the conical special Kahler domain Mα in the affine special coordinates qa =

(xI , yJ). The first two terms of (6.1) are manifestly invariant under symplectic

transformations A ∈ Sp(2n + 2, R) since the 1-form∑

paΩabdpb is invariant.

The invariance of the last term is stated in the next lemma establishing the

existence of the metric gN . The proof of the lemma will be given in the next

section together with a geometric interpretation.

Lemma 4 The tensors H−1α and H−1

β defined on Mα and Mβ are related by

H−1α = AαβH−1

β ATαβ

on overlaps Mα ∩ Mβ.

The metric gN is locally a quaternionic Kahler metric. To see that the local

quaternionic structures are consistent, we observe that the coordinate transfor-

mations Nβ → Nα are orientation preserving isometries and that an orientation

preserving isometry between two quaternionic Kahler manifolds of nonzero scalar

curvature automatically maps the quaternionic structures to each other. This fol-

lows from the fact that the restricted holonomy group of a quaternionic Kahler

manifold of nonzero scalar curvature together with the orientation uniquely de-

termines the quaternionic structure. (Notice that the orientation is needed, since

the symmetric quaternionic Kahler manifold

SO0(4, n)SO(4) × SO(n)


admits precisely two parallel skew-symmetric quaternionic structures, which,

however, induce opposite orientations.) ⊓⊔

The correspondence (M, gM ) 7→ (N, gN ) established in Theorem 9 is a global

version of the c-map of Ferrara and Sabharwal. We will still call it the supergravity

c-map.

Theorem 10 The supergravity c-map maps (isomorphism classes of) complete

projective special Kahler manifolds (M, gM ) of dimension 2n to (isomorphism

classes of) complete quaternionic Kahler manifolds (N, gN ) of dimension 4n+4

of negative scalar curvature such that N is a vector bundle over M with totally

geodesic zero section isometric to M .

Proof: This follows from Theorem 9 and Theorem 3. ⊓⊔

6.1. From Griffiths to Weil flags in special Kahler geometry. Let us consider

the complex vector space V = C2n+2 = R2n+2 ⊗C with its standard symplectic

structure Ω =∑

dzI ∧ dwI and pseudo-Hermitian sesquilinear metric

γ(u, v) =√−1Ω(u, v), u, v ∈ V,

of split signature. We denote by Grk,l0 (V ) the Grassmannian of complex La-

grangian subspaces of signature (k, l), where k + l = n + 1. For k ≥ 1, let

F k,l0 (V ) denote the complex manifold of flags (ℓ, L), where L ∈ Grk,l

0 (V ) and

ℓ ⊂ L is a positive definite line. Notice that we have a canonical holomorphic

projection

F k,l0 (V ) =

Sp(R2n+2)U(1) × U(k − 1, l)

−→ Grk,l0 (V ) =

Sp(R2n+2)U(k, l)

, (ℓ, L) 7→ L,

which is Sp(R2n+2)-equivariant.


Proposition 2 There exists a canonical Sp(R2n+2)-equivariant diffeomor-

phism

ψ : F k,l0 (V ) −→ F l+1,k−1

0 (V ).

Proof: For (ℓ, L) ∈ F k,l0 (V ) we put

E := v ∈ L|v ⊥ ℓ

and define ψ(ℓ, L) := (ℓ, L′) where

L′ := ℓ + E.

⊓⊔

In particular, we obtain an equivariant diffeomorphism from the manifold of

Griffiths flags to the manifold of Weil flags:

ψ : F 1,n0 (V ) −→ Fn+1,0

0 (V ).

Remark: In order to motivate the terminology we observe that given L ∈

F 1,n0 (V ) and a lattice Γ ⊂ R2n+2 the quotient of W = V/L by (the image of) Γ

is a complex torus which is analogous to the Griffiths intermediate Jacobian

H3(X, C)H3,0(X, C) + H2,1(X, C) + H3(X, Z)

whereas the quotient of W ′ = V/L′ by Γ is analogous to the Weil intermediate

Jacobian

H3(X, C)H3,0(X, C) + H1,2(X, C) + H3(X, Z)

associated to the Hodge structure of a Calabi-Yau 3-fold X. It is known that

the bundle of Griffiths intermediate Jacobians over the (conical special Kahler)

deformation space MX = (J, ν) of complex structures J of X gauged by a

J-holomorphic volume form ν carries a hyper-Kahler metric obtained from the


affine version of the c-map [10]. Similarly, a certain bundle of Weil intermediate

Jacobians is quaternionic Kahler by the supergravity c-map [29,38].

Recall that there is a totally geodesic Sp(R2n+2)-equivariant embedding

ι : Grk,l0 (V ) =

Sp(R2n+2)U(k, l)

−→ Sym12k,2l(R2n+2) =

SL(2n + 2, R)SO(2k, 2l)

(6.4)

into the space Sym12k,2l(R2n+2) of symmetric unimodular matrices of signature

(2k, 2l). The embedding ι is described geometrically in [14]. In the next lemma

we give an explicit description of ι in terms of coordinates. Local holomorphic co-

ordinates near a point L0 ∈ Grk,l0 (V ) can be described as follows. Since Grk,l

0 (V )

is a homogeneous space we can assume that

L0 = span

∂

∂z0+ i

∂

∂w0, . . . ,

∂

∂zk−1+ i

∂

∂wk−1,

∂

∂zk− i

∂

∂wk, . . . ,

∂

∂zn− i

∂

∂wn

.

An open neighbourhood U of L0 is given by

U := L ∈ Grk,l0 (V )|L ∩ (Cn+1)∗ = 0,

where

(Cn+1)∗ = (z, w) ∈ V |z = 0 ⊂ V = T ∗Cn+1 = Cn+1 ⊕ (Cn+1)∗.

We will now explain that any point L ∈ U is described by a complex symmetric

matrix S = (SIJ) such that the real matrix ImSIJ has signature (k, l). Let

us denote by Symk,l(Cn+1) the complex vector space of all such matrices. Any

S ∈ Symk,l(Cn+1) defines a Lagrangian subspace

L = L(S) = (z, w) ∈ V |wI =∑

SIJzJ ⊂ V

and one can easily check that the map S 7→ L(S) is a biholomorphism Symk,l(Cn+1) →

U ⊂ Grk,l0 (V ). Notice that L0 = L(S0), S0 = iIk,l = idiag(1k,−1l). It is well


know that the matrix S transforms as

S 7→ S′ = (C + DS)(A + BS)−1 (6.5)

under a symplectic transformation

O =

A B

C D

. (6.6)

Lemma 5 The restriction of the map (6.4) to the the open subset U ⊂

Grk,l0 (V ) is given by

ι|U : U ∼= Symk,l(Cn+1) → Sym12k,2l(R2n+2),

S = R + iI 7→ ι(L(S)) = gS = (gSab) :=


−I−1R I−1

.(6.7)

Proof: The above formula shows that gS0 = diag(Ik,l, Ik,l). To prove that gS =

ι(L(S)) for all S ∈ U it suffices to check that

gS′= OT,−1gSO−1

for S′ defined in (6.5). This follows from Lemma 6 which is stated and proved

below. ⊓⊔

The relation between the manifolds of Griffiths and Weil flags, the associated

Grassmannians, and spaces of symmetric matrices is summarized in the following

diagram.

F 1,n0 (V )

ψ//

²²

Fn+1,00 (V )

²²

Gr1,n0 (V )Ä _

ι

²²

Grn+1,00 (V )Ä _

ι

²²

Sym12,2n(R2n+2) Sym1

2n+2,0(R2n+2)

(6.8)


Lemma 6 Let N = R + iI be a complex symmetric (n + 1) × (n + 1) matrix

with invertible imaginary part. Using the decomposition into real and imaginary

part, define the real symmetric (2n + 2) × (2n + 2) matrix

H = (Hab) =


−I−1R I−1

.

Then H is invertible with inverse matrix

H−1 = (Hab) =

I−1 I−1R

RI−1 I + RI−1R

. (6.9)

Moreover, N transforms fractionally linearly under symplectic transformations

(6.6),

N → (C + DN)(A + BN)−1

if and only if H, and, hence H−1 transform as tensors:

H → OT,−1HO−1 , H−1 → OH−1OT .

Remarks: This lemma relates the transformation properties of vector multiplet

couplings in special holomorphic and special real coordinates. While we need the

lemma to establish the well-definiteness of the local c-map, it applies to the rigid

c-map as well. In the rigid case the role of N is played by two times the matrix

of second derivatives of the holomorphic prepotential, 2(FIJ ), while the role of

H is played by the Hessian metric ∂2H, where H is the Legendre transform of

two times the imaginary part of the holomorphic prepotential. When passing to

supergravity, 2(FIJ) is replaced by N, while the Hessian metric is replaced by H.

When using the superconformal calculus to construct vector multiplet couplings

these replacements are induced by eliminating certain auxiliary fields. From a

geometrical perspective these replacements can be understood as follows. The


kinetic terms of both scalar fields and vector fields must be positive definite in

a physically acceptable theory. In a theory of rigid supermultiplets, the relevant

coupling matrix for both types of fields is the metric 2ImFIJ of the affine spe-

cial Kahler manifold, which therefore must be positive definite. In the locally

supersymmetric theory scalar and vector fields have different couplings matri-

ces. The coupling matrix for the scalars is the metric g of the projective special

Kahler manifold, while the coupling matrix for the vector fields is N, with the

kinetic terms given by the imaginary part I. Therefore g and I must be positive

definite, which is equivalent to imposing that the corresponding conical affine

special Kahler metric has complex Lorentz signature.

Proof: We now prove Lemma 6. It is trivial to verify that (6.9) is the inverse

matrix of H. Note that I is invertible by assumption. The relation between the

transformation properties of N and H can be verified by direct calculation. Such

calculations have occured in the supergravity literature, see for example [16], so

that we only need to indicate the main steps. Let N′ = R′ + iI′ be the matrix

obtained by fractionally linear action of the symplectic matrix O on N. This

is equivalent to H transforming as a tensor if and only if the following three

relations hold

(I + RI−1R)′ = D(I + RI−1R)DT + DRI−1CT + CI−1RDT

+CI−1CT , (6.10)

(RI−1)′ = D(I + RI−1R)BT + DRI−1AT + CI−1RBT

+CI−1AT , (6.11)

(I−1)′ = B(I + RI−1R)BT + BRI−1AT + AI−1RBT

+AI−1BT . (6.12)


The matrices A, B,C,D are block sub-matrices of the symplectic matrix O,

which satisfies

OT ΩO = Ω , where Ω =

0 1n+1

−1n+1 0

.

Therefore they satisfy

AT C = CT A , BT D = DT B , AT D − CT B = 1 .

This can be used to verify the following the useful identities

UT (C + DN) = (C + DN)T U , UT (C + DN) = (C + DN)T U − 2iI , (6.13)

where U = U(N) := A + BN. Two further identities are obtained by complex

conjugation. By repeated use of these identities, we can show that

2iI = N − N = UT (C + DN)U−1U − UT (C + DN) . (6.14)

The imaginary part I of N transforms under symplectic transformations into

I′ = − i

2[(C + DN)U−1 − (C + DN)U−1] .

Using identity (6.14) this can be rewritten as

I′ = U−1,T IU−1 = U−1,T IU−1 ,

where the second equation holds because I is real. Since I is invertible by as-

sumption, we conclude

(I−1)′ = UI−1UT = UI−1UT . (6.15)

Writing out U = A + BN and N = R + iI we obtain (6.12). Next, we note that

the real part R of N transforms into

R′ =12[(C + DN)U−1 + (C + DN)U−1] .


Combining this with (6.15) we obtain

(RI−1)′ =12[(C + DN)I−1UT + (C + DN)−1UT ] .

Expressing U,N in terms of A,B,C,D and R, I, we obtain (6.11). Finally, we

multiply (RI−1)′ by R′ from the right and add I′. After repeated use of the

identities (6.13) we finally obtain (6.12). ⊓⊔

Let L ⊂ V = C2n+2 be a Lagrangian subspace defined by S ∈ Sym1,n(Cn+1)

and ℓ = C(z, w)T ⊂ L. Then w = Sz, (ℓ, L) ∈ F 1,n0 (V ) and (ℓ, L′) = ψ(ℓ, L) ∈

Fn+1,00 (V ), cf. (6.8).

Lemma 7 The positive definite Lagrangian subspace L′ ⊂ V corresponds to

the following matrix S′ ∈ Symn+1,0(Cn+1):

S′IJ := SIJ + i

∑K NIKzK

∑L NJLzL∑

IJ NIJzIzJ, NIJ := 2Im SIJ . (6.16)

Proof: The following calculation shows that ℓ = C(z, w)T is contained in the

Lagrangian subspace L(S′) defined by S′:

S′z = Sz + 2i(ImS)z = Sz.

Next we consider the orthogonal complement E of the line ℓ in L. A vector

(u, Su)T ∈ L belongs to E if and only if∑

NIJzI uJ = 0. In that case we obtain

S′u = Su = Su,

which proves that E is contained in L(S′). Therefore, L(S′) = L′. ⊓⊔

Corollary 5

(i) The complex matrix S′ = N ∈ Symn+1,0(Cn+1) occurring in the definition

of the Ferrara-Sabharwal metric, see (4.3), is related to the matrix S = FIJ ∈

Sym1,n(Cn+1) by the correspondence S 7→ S′ of the previous lemma, which is

induced by the map ψ from Griffiths flags to Weil flags, cf. (6.17).


(ii) The real matrix (6.3) occurring in the formula (6.1) for the Ferrara-

Sabharwal metric is given by

H = gS′= gN,

where the map S 7→ gS is defined in (6.7).

The following diagram gives an overview of the relations between the La-

grangian subspaces L = L(S), L′ = L(S′) and the corresponding real matrices

gS = (gab) and gS′= Hab:

(ℓ, L) ∈ F 1,n0 (V )

ψ//

²²

Fn+1,00 (V ) ∋ (ℓ, L′)

²²

L ∈ Gr1,n0 (V )Ä _

ι

²²

Grn+1,00 (V ) ∋ L′

Ä _

ι

²²

gS ∈ Sym12,2n(R2n+2) Sym1

2n+2,0(R2n+2) ∋ gS′,

(6.17)

where the line ℓ is generated by the vector (z, Sz) = (z, S′z).

Corollary 6 The (indefinite) affine special Kahler metric g = 2∑

gabdqadqb

is related to the positive definite metric g′ = 2∑

Habdqadqb by

g′|D = g|D, g′|D⊥ = −g|D⊥ ,

where D is defined in Definition 3 (iv).

Proof: This follows from the geometric description of the map ι : L = L(S) 7→ gS

[14]. Recall that in the affine special coordinates qa = (xI = Re zI , yJ = Re wJ)

the Kahler form is given by ω = 2∑

dxi ∧ dyi. The matrix g = gS represents

the scalar product Re γ|L in the Darboux coordinates√

2qa restricted to L. We

can compare the restrictions of γ to L and L′ by the following isomorphism of


real vector spaces Ψ : L → L′:

Ψ |ℓ := Id, Ψ(v) := v for all v ∈ E.

We can easily see that Ψ is an isometry on ℓ = L ∩ L′ and an antiisometry

E → E on E. In fact, γ(v, v) = −γ(v, v) for all v ∈ V . This shows that the

metric Ψ∗Re γ|L′ is related to g = Re γ|L by changing the sign on the orthog-

onal complement of ℓ. Finally, the Gram matrix gS′= (Hab) of Re γ|L′ in the

coordinates√

2qa|L′ is the same as that of Ψ∗Re γ|L′ in the coordinates√

2qa|L,

since qa Ψ = qa. ⊓⊔

We now finally prove Lemma 4 and thus complete the proof of Theorem 9.

Proof: Let us denote by Fα the holomorphic prepotential of the special Kahler

domain Mα and put Sα := FαIJ . We know that the complex matrices Sα and

Sβ are related by a fractional linear transformation associated with a symplec-

tic transformation O = Aαβ ∈ Sp(R2n+2), which relates the Gram matrices gSα

and gSβ of the corresponding affine special Kahler metrics gMαand gMβ

in affine

Darboux coordinates. Since all the maps in the diagram (6.17) are Sp(R2n+2)-

equivariant, this implies that the matrices Nα and Nβ are related by the same

fractional linear transformation and that gS′α and gS′

β are related by the sym-

plectic transformation O. This shows that Hα transforms as claimed in Lemma

4. ⊓⊔

Acknowledgements. This work was supported by the German Science Foundation (DFG) un-

der the Collaborative Research Center (SFB) 676. The work of T.M. was supported in part

by STFC grant ST/G00062X/1.


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